# What is the greatest common monomial factor of 2k^3 + 6k^2 - 14k?

May 8, 2015

The answer is $2 k \left({k}^{2} + 3 k - 7\right)$, where $2 k$ is the greatest common monomial factor.

To begin with this problem, let's consider the context of what the problem is asking. It wants us to find the common monomial factor of the quadratic. What this means is how can it be factored out into an expression that still acts as the original function, but in a way it can be done much easier in simplification.

In each term, we notice that $2$, $3$, and $14$ are all divisible by two. In addition, each term has a $k$ variable that can be factored out as well (following a similar division rule). The following link helps conceptually see it:

In numerical steps:

$2 {k}^{3} + 6 {k}^{2} - 14 k \implies$factor out a $2$ and divide each term by two as well.

$2 \left({k}^{3} + 3 {k}^{2} - 7 k\right) \implies$factor out a $k$ variable and divide the rest of the terms by $k$,

which then becomes $2 k \left({k}^{2} + 3 k - 7\right)$. The greatest common factor is $2 k$ because, according to our factored equation, it is most commonly factored out for all the terms in the original polynomial equation.

This is really useful when you are dividing/multiplying expressions; by doing these kinds of factors, you can make equations/answers much simpler if they can be. Here is a good video on factoring quadratic equations and simplification from Mark Lehain: